From cc77bcc7fc6d75293ecf5791aad75d7d11be0287 Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Thu, 18 Jul 2024 17:01:51 +0200 Subject: [PATCH] rapport : changing margins, explaining ARI negatives, adding NA robustness sims, --- rapport/chapter4-simulations/inference.tex | 8 +- .../chapter4-simulations/na-robustness.tex | 41 +- rapport/rapport.pdf | Bin 774335 -> 813023 bytes rapport/rapport.tex | 1 + .../na_robustness/ari-dim-model.tex | 3922 +++++++++++++++++ tikz/simulations/na_robustness/auc-model.tex | 2021 +++++++++ 6 files changed, 5989 insertions(+), 4 deletions(-) create mode 100644 tikz/simulations/na_robustness/ari-dim-model.tex create mode 100644 tikz/simulations/na_robustness/auc-model.tex diff --git a/rapport/chapter4-simulations/inference.tex b/rapport/chapter4-simulations/inference.tex index 8582a52..93a85f4 100644 --- a/rapport/chapter4-simulations/inference.tex +++ b/rapport/chapter4-simulations/inference.tex @@ -70,8 +70,12 @@ use the following indicators: their true values. ($Q_1 = 4$ and $Q_2 = 4$) \item Finally, we assess the quality of the node grouping by computing the Adjusted Rand Index \parencite{hubertComparingPartitions1985}, ARI = 0 - for a random grouping, ARI = 1 for a perfect recovery. For each - network, for the + for a random grouping, ARI = 1 for a + perfect match between groupings\footnote{Please note that even if Rand + Index can only yield values between 0 and 1, ARI can return + negative values if the RI is less than the expected value. This + indicates a structure in grouping discordance.}. + For each network, for the $\pi\text{-}colBiSBM$, $\rho\text{-}colBiSBM$, $\pi\rho\text{-}colBiSBM$ we compare the inferred block memberships to the real ones by computing the mean of the ARI per axis over the two diff --git a/rapport/chapter4-simulations/na-robustness.tex b/rapport/chapter4-simulations/na-robustness.tex index f9ffe98..ec9d949 100644 --- a/rapport/chapter4-simulations/na-robustness.tex +++ b/rapport/chapter4-simulations/na-robustness.tex @@ -36,11 +36,15 @@ less with the other communities. And $\bm{\alpha}^{nested}$ represents a common structure detected in ecology with generalist and specialist species and a \enquote{nested} structure. -The collections contain two networks of size $n^{m=1}_1 = n^{m=1}_2 = 40$ and +The collections contain two networks ($M=2$) of size $n^{m=1}_1 = + n^{m=1}_2 = 40$ and $n^{m=2}_1 = n^{m=2}_2 = 120$. One collection is generated for each $colBiSBM$ model. And the nodes block memberships (i.e., the row and column blocks they belong to) are saved. +Per $colBiSBM$ model, 10 collections are generated and their results are +averaged. + In the network $m=1$ (i.e., the smaller one) a proportion of the edges $p_{\texttt{NA}}$ see their values replaced by \texttt{NA}s, the \enquote{forgotten} values are stored. @@ -57,6 +61,39 @@ and we store the same predictions. \emph{Area Under the Curve} (AUC) for predicted versus real link values and the ARI for predicted versus real block memberships. +For the comparison we subtract the metric given by the LBM to the one +given by $colBiSBM$ and denote it $\Delta\mbox{metric}$. + +\begin{figure}[ht] + \centering + \input{../tikz/simulations/na_robustness/auc-model} + \caption{$\Delta\mbox{AUC}$ in function of $p_{\texttt{NA}}$. The dashed red + lines indicate the value 0 for which + $\mbox{AUC}_{LBM} = \mbox{AUC}_{colBiSBM}$} + \label{fig:auc-plot} +\end{figure} + +\begin{figure}[ht] + \centering + \input{../tikz/simulations/na_robustness/ari-dim-model} + \caption{$\Delta\mbox{ARI}$ in function of $p_{\texttt{NA}}$. The dashed red + lines indicate the value 0 for which + $\mbox{ARI}_{LBM} = \mbox{ARI}_{colBiSBM}$} + \label{fig:ari-dim-plot-na} +\end{figure} + + \paragraph{Results} +On figure~\ref{fig:auc-plot} one can see that overall the nested structure seems +to be the one benefitting most from the collection model having generally +slightly higher $\Delta$AUC than the modular one. +But in general it seems that for $\epsilon\in[0.1,0.7]$ there are no clear +differences between LBM and colBiSBM regarding link prediction. For $\epsilon + \in[0.8,0.9]$ this is where the collection model seems to be most effective. - +For the ARI, figure~\ref{fig:ari-dim-plot-na} suggests that collection model +does at least as well as LBM and improves nodes memberships recovery for modular +structure starting from $\epsilon = 0.7$. 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+\end{scope} +\end{tikzpicture} diff --git a/tikz/simulations/na_robustness/auc-model.tex b/tikz/simulations/na_robustness/auc-model.tex new file mode 100644 index 0000000..30409a5 --- /dev/null +++ b/tikz/simulations/na_robustness/auc-model.tex @@ -0,0 +1,2021 @@ +% Created by tikzDevice version 0.12.6 on 2024-07-18 16:43:28 +% !TEX encoding = UTF-8 Unicode +\begin{tikzpicture}[x=1pt,y=1pt] +\definecolor{fillColor}{RGB}{255,255,255} +\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (361.35,180.67); +\begin{scope} +\path[clip] ( 37.09,105.68) rectangle (286.68,175.17); +\definecolor{drawColor}{gray}{0.92} + +\path[draw=drawColor,line width= 0.3pt,line join=round] ( 37.09,114.06) -- + (286.68,114.06); + +\path[draw=drawColor,line width= 0.3pt,line join=round] ( 37.09,126.62) -- + (286.68,126.62); + +\path[draw=drawColor,line width= 0.3pt,line join=round] ( 37.09,139.19) -- + (286.68,139.19); + +\path[draw=drawColor,line width= 0.3pt,line join=round] ( 37.09,151.75) -- + (286.68,151.75); 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